Probability: Fluency and Exam Drill (Part 2) Worksheet with Answers

Revision Library

revisionlibrary.co.uk

GCSE · Statistics and Probability

2.12DB Probability: Fluency and Exam Drill (Part 2)

EDEXCEL 1MA1 · Calculator allowed · about 90 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

1

Some probability questions about everyday events.

(a)A fair coin is flipped once. Write down the probability, as a fraction, that it lands on tails.(1)

(b)A standard, well-shuffled pack of 52 playing cards is used. One card is picked at random. Write down the probability that the card is a picture card (a Jack, Queen or King).(1)

(c)A fair spinner has 4 equal sections numbered 1, 2, 3, 4. The spinner is spun once. Write down the probability that it lands on a number greater than 4.(1)

(Total for Question 1 is 3 marks)

2

A charity raffle uses 45 tickets: 21 are blue, 15 are yellow and 9 are pink. One ticket is drawn from the raffle at random.

(a)Write down the probability that the ticket is yellow.(1)

(b)Write down the probability that the ticket is not blue.(1)

(c)Write down the probability that the ticket is blue or pink.(1)

(Total for Question 2 is 3 marks)

3

A school uniform shop sells polo shirts in 5 colours (white, navy, green, red, black) and caps in 3 colours (grey, blue, orange). A customer buys one polo shirt and one cap.

(a)List all the possible polo-shirt-and-cap colour combinations a customer could buy.(2)

(b)A customer buys a random combination. Find the probability that they buy a red shirt and a blue cap.(1)

(c)Find the probability that the cap is grey or orange.(1)

(Total for Question 3 is 4 marks)

4

A fair spinner has 12 equal sections, numbered 1 to 12. The spinner is spun once.

(a)Find the probability that the spinner lands on a multiple of 4.(1)

(b)Find the probability that the spinner lands on a factor of 12.(1)

(c)Find the probability that the spinner lands on a square number.(1)

(Total for Question 4 is 3 marks)

5

A cafe surveyed 70 diners on one lunchtime about whether they ordered a hot drink.

Hot drink No hot drink Total
Adults 32 ? 50
Children ? 14 ?
Total ? ? 70

Some values in the table are missing (shown as ?).

(a)Complete the two-way table.(3)

(b)One diner is picked at random from the 70. Find the probability that the diner is a child who ordered a hot drink.(1)

(c)Given that a diner picked at random did not order a hot drink, find the probability that they are a child.(2)

(Total for Question 5 is 6 marks)

6

On a particular day, the probability of rain is 3/10 and the probability of it being cloudy (with no rain) is 2/5. The day can only be rainy, cloudy or sunny.

(a)Work out the probability that the day is sunny.(1)

(b)Work out the probability that the day is either rainy or sunny (but not cloudy).(1)

(Total for Question 6 is 2 marks)

7

A jar contains only yellow, purple and orange sweets, in the ratio yellow : purple : orange = 2 : 5 : 3.

(a)Find P(purple), the probability that a sweet taken at random from the jar is purple.(2)

(b)Find P(orange).(1)

(c)There are 80 sweets in the jar in total. Work out how many of the sweets are yellow.(2)

(Total for Question 7 is 5 marks)

8

Spinner A has four equal sections numbered 1, 2, 3, 4. Spinner B has three equal sections numbered 1, 2, 3. Both spinners are spun once and the two scores are multiplied together to give a total score.

(a)Complete a sample space diagram to show all 12 possible product scores.(2)

(b)Find the probability that the product is even.(1)

(c)Find the probability that the product is greater than 6.(2)

(Total for Question 8 is 5 marks)

9

Chidi rolls a six-sided dice 60 times and scores a six 14 times. Aisha rolls an identical dice 500 times and scores a six 79 times.

(a)Work out the relative frequency of scoring a six in each experiment, giving your answers as decimals to 3 significant figures.(2)

(b)State, with a reason, whose experiment gives the more reliable estimate of the probability of scoring a six.(1)

(c)Kwame says: 'Chidi's dice must be biased, because his relative frequency of 14 out of 60 is higher than Aisha's.' Explain why Kwame's conclusion is not necessarily correct.(2)

(Total for Question 9 is 5 marks)

10

A greengrocer checks a random sample of 50 apples from a large delivery of 2000 apples and finds that 4 are bruised.

(a)Estimate the probability that an apple from the delivery is bruised.(1)

(b)Use your answer to part (a) to estimate the number of apples in the delivery that are NOT bruised.(2)

(c)The greengrocer then checks a second random sample of 40 apples from the same delivery and finds that 2 are bruised. Use this second sample to find another estimate of the probability that an apple is bruised, and comment on what this suggests about the estimate found in part (a).(2)

(Total for Question 10 is 5 marks)

11

60 students were asked whether they play chess (C) or draughts (D). 34 students play chess, 28 play draughts, and 9 play neither game.

Figure (to be drawn): A rectangle representing all 60 students in the survey, containing two overlapping circles: the left circle labelled C (Chess) and the right circle labelled D (Draughts). All four regions (C only, the overlap between C and D, D only, and the region outside both circles, inside the rectangle) are left blank for the student to fill in with the correct numbers.

(a)Find n(C and D), the number of students who play both chess and draughts.(2)

(b)One student is selected at random from the 60. Find the probability that they play exactly one of the two games.(1)

(c)Given that a randomly selected student plays chess, find the probability that they also play draughts.(2)

(Total for Question 11 is 5 marks)

12

200 runners entered a race. 120 of the runners were male and the rest were female. 0.75 of the male runners finished the race, and 0.6 of the female runners finished the race.

Figure (to be drawn): A two-stage frequency tree diagram. The first branch splits the 200 runners into Male and Female. Each of those two branches then splits again into Finished and Did not finish, with a blank box at the end of each of the four final branches for the student to fill in the frequency.

(a)Complete the frequency tree to show the number of runners in each category.(3)

(b)Find the probability that a runner selected at random did not finish the race.(1)

(Total for Question 12 is 4 marks)

13

A fair 20-sided dice is numbered 1 to 20 and rolled once.

(a)Find the probability that the score is a multiple of 3 or a multiple of 5.(3)

(b)Explain why the answer to part (a) is not simply P(multiple of 3) + P(multiple of 5).(1)

(Total for Question 13 is 4 marks)

14

Priya is working out the probability of rolling at least one 6 when a fair six-sided dice is rolled twice. Her working is shown below:

P(at least one 6) = P(6) + P(6) = 1/6 + 1/6 = 2/6 = 1/3

(a)Identify the error in Priya's method.(1)

(b)Work out the correct probability that at least one 6 is rolled.(3)

(Total for Question 14 is 4 marks)

15

A drawer contains 12 ties: 5 are patterned and 7 are plain. Wilf takes two ties from the drawer at random, one after the other, without replacement.

(a)Work out the probability that both ties are plain.(2)

(b)Work out the probability that at least one of the two ties is patterned.(3)

(Total for Question 15 is 5 marks)

16

A company employs 120 staff across two departments. The table shows some information about how many staff in each department work from home.

Works from home Does not Total
Sales 30 ? 50
Admin ? 55 ?
Total ? ? 120

Some values in the table are missing (shown as ?).

(a)Complete the two-way table.(2)

(b)Find the probability that a randomly selected employee works from home.(1)

(c)By comparing suitable probabilities, determine whether Sales or Admin has the greater proportion of staff working from home. Justify your answer.(2)

(Total for Question 16 is 5 marks)

17

The probability that Freya scores a goal from a penalty is 0.7, independently each time she takes one. Freya takes three penalties.

(a)Work out the probability that she scores on her first two penalties but misses her third.(2)

(b)Work out the probability that she scores at least two of her first three penalties.(3)

(Total for Question 17 is 5 marks)

18

A pencil case contains n black pens and 4 red pens, and no other pens. Two pens are taken at random from the pencil case, one after the other, without replacement. The probability that both pens are black is 1/3.

(a)Show that n² - 5n - 6 = 0.(3)

(b)Hence find the value of n.(2)

(Total for Question 18 is 5 marks)

19

A box contains 3 fair coins and 1 biased coin. The biased coin lands on heads with probability 0.9. Each fair coin lands on heads with probability 0.5. A coin is chosen at random from the box and flipped once.

(a)Find the probability that the coin lands on heads.(3)

(b)Given that the coin landed on heads, find the probability that it was the biased coin.(3)

(Total for Question 19 is 6 marks)

Mark scheme · 2.12DB Probability: Fluency and Exam Drill (Part 2)

Question 1

(a) B1 1/2 oe
(a) Answer: 1/2
(b) B1 12/52 oe (3/13)
(b) Answer: 3/13
(c) B1 0 cao
(c) Answer: 0

Question 2

(a) B1 15/45 oe (1/3)
(a) Answer: 1/3
(b) B1 24/45 oe (8/15)
(b) Answer: 8/15
(c) B1 30/45 oe (2/3)
(c) Answer: 2/3

Question 3

(a) M1 at least 10 of the 15 combinations listed correctly, in a systematic order
(a) A1 all 15 combinations listed correctly, oe
(a) Answer: white-grey, white-blue, white-orange, navy-grey, navy-blue, navy-orange, green-grey, green-blue, green-orange, red-grey, red-blue, red-orange, black-grey, black-blue, black-orange
(b) B1 1/15 ft from part (a)
(b) Answer: 1/15
(c) B1 10/15 oe (2/3), ft from part (a)
(c) Answer: 2/3

Question 4

(a) B1 3/12 oe (1/4)
(a) Answer: 1/4
(b) B1 6/12 oe (1/2)
(b) Answer: 1/2
(c) B1 3/12 oe (1/4)
(c) Answer: 1/4

Question 5

(a) B1 Adults with no hot drink = 18 (50 - 32)
(a) B1 Children total = 20 (70 - 50) and children with a hot drink = 6 (20 - 14)
(a) B1 Total with a hot drink = 38 (32 + 6) and total with no hot drink = 32 (18 + 14)
(a) Answer: Adults no-hot-drink = 18; Children total = 20, Children hot-drink = 6; Total hot-drink = 38, Total no-hot-drink = 32
(b) B1 6/70 oe (3/35), ft from part (a)
(b) Answer: 3/35
(c) M1 restricts the sample space to the 32 diners with no hot drink, ft from part (a)
(c) A1 14/32 oe (7/16)
(c) Answer: 7/16

Question 6

(a) B1 3/10 cao
(a) Answer: 3/10
(b) B1 3/5 oe (6/10), ft from part (a)
(b) Answer: 3/5

Question 7

(a) M1 total ratio parts = 2 + 5 + 3 = 10 oe (5/10 unsimplified)
(a) A1 1/2 cao
(a) Answer: 1/2
(b) B1 3/10 cao
(b) Answer: 3/10
(c) M1 P(yellow) = 1 - 1/2 - 3/10 (= 1/5), ft from parts (a) and (b)
(c) A1 16 cao
(c) Answer: 16

Question 8

(a) M1 at least 8 of the 12 products correct
(a) A1 all 12 products correct: 1,2,3 / 2,4,6 / 3,6,9 / 4,8,12
(a) Answer: products for A=1: 1,2,3; A=2: 2,4,6; A=3: 3,6,9; A=4: 4,8,12
(b) B1 8/12 oe (2/3), ft from part (a)
(b) Answer: 2/3
(c) M1 identifies the products greater than 6: 9, 8, 12
(c) A1 3/12 oe (1/4), ft from part (a)
(c) Answer: 1/4

Question 9

(a) B1 Chidi: 14/60 = 0.233 (awrt)
(a) B1 Aisha: 79/500 = 0.158
(a) Answer: Chidi: awrt 0.233; Aisha: 0.158
(b) B1 Aisha's, because a larger number of trials generally gives a more reliable estimate of the true probability, oe
(b) Answer: Aisha's experiment
(c) M1 recognises that relative frequency naturally varies between different sets of trials, particularly for a small number of trials, oe
(c) A1 concludes that Chidi's smaller sample alone is not enough evidence to prove bias; more trials (or comparing to the theoretical probability 1/6) would be needed, oe
(c) Answer: Kwame is not necessarily correct: relative frequency varies between trials, especially for a small sample like Chidi's 60 rolls, so a higher result there does not on its own prove the dice is biased

Question 10

(a) B1 4/50 oe (2/25, 0.08)
(a) Answer: 2/25
(b) M1 1 - 2/25 (= 23/25) x 2000, ft from part (a)
(b) A1 1840 cao
(b) Answer: 1840
(c) M1 2/40 = 0.05 found
(c) A1 valid comparison-based comment, e.g. the two samples give different estimates (0.08 and 0.05), so neither sample alone is fully reliable and a larger combined sample would give a more trustworthy estimate, oe
(c) Answer: The second sample gives an estimate of 0.05, which is lower than the 0.08 from part (a); the two estimates disagree, showing that a sample of 50 (or 40) may not be large enough to give a fully reliable estimate

Question 11

(a) M1 34 + 28 - (60 - 9) oe
(a) A1 11 cao
(a) Answer: 11
(b) B1 40/60 oe (2/3), ft from part (a)
(b) Answer: 2/3
(c) M1 restricts the sample space to the 34 students who play chess, ft from part (a)
(c) A1 11/34 oe
(c) Answer: 11/34

Question 12

(a) M1 male finished = 0.75 x 120 = 90 (and male did not finish = 30)
(a) M1 female = 200 - 120 = 80; female finished = 0.6 x 80 = 48 (and female did not finish = 32)
(a) A1 all four frequencies correct: 90, 30, 48, 32
(a) Answer: Male finished = 90, male did not finish = 30; Female finished = 48, female did not finish = 32
(b) B1 62/200 oe (31/100, 0.31), ft from part (a)
(b) Answer: 31/100

Question 13

(a) M1 identifies the multiples of 3 (3, 6, 9, 12, 15, 18: 6 values) and the multiples of 5 (5, 10, 15, 20: 4 values)
(a) M1 recognises 15 is a multiple of both and subtracts the overlap: 6 + 4 - 1
(a) A1 9/20 cao
(a) Answer: 9/20
(b) B1 because 15 is a multiple of both 3 and 5, so simply adding the two probabilities would count the score of 15 twice, oe
(b) Answer: Adding P(multiple of 3) and P(multiple of 5) directly would count the score 15 (which is a multiple of both) twice, so the overlap must be subtracted once

Question 14

(a) B1 simply adding the two single-roll probabilities is not a valid method for 'at least one' across two rolls, because it does not correctly account for the outcome where a 6 occurs on both rolls, oe
(a) Answer: Priya cannot find P(at least one 6) by adding the two single-roll probabilities; that method does not correctly handle the outcome where both rolls are 6
(b) M1 P(no six on either roll) = 5/6 x 5/6 (= 25/36)
(b) M1 1 - 25/36
(b) A1 11/36 cao
(b) Answer: 11/36

Question 15

(a) M1 7/12 x 6/11
(a) A1 42/132 oe (7/22)
(a) Answer: 7/22
(b) M1 P(both plain) = 7/22, ft from part (a)
(b) M1 1 - 7/22
(b) A1 15/22 cao
(b) Answer: 15/22

Question 16

(a) M1 Sales does not = 20 (50 - 30); Admin total = 70 (120 - 50)
(a) A1 Admin works from home = 15 (70 - 55); Total works from home = 45 (30 + 15), total does not = 75 (20 + 55)
(a) Answer: Sales does-not = 20; Admin total = 70, Admin works-from-home = 15; Total works-from-home = 45, total does-not = 75
(b) B1 45/120 oe (3/8), ft from part (a)
(b) Answer: 3/8
(c) M1 finds P(works from home given Sales) = 30/50 (= 0.6) and P(works from home given Admin) = 15/70 (awrt 0.214), ft from part (a)
(c) A1 concludes Sales has the greater proportion, since 0.6 is greater than 0.214, cao
(c) Answer: Sales has the greater proportion working from home: 30/50 = 0.6 compared with 15/70 (awrt 0.214) for Admin

Question 17

(a) M1 0.7 x 0.7 x 0.3
(a) A1 0.147 cao
(a) Answer: 0.147
(b) M1 P(exactly two scored) = 3 x 0.7² x 0.3 (= 0.441), recognising the three possible orders
(b) M1 P(exactly three scored) = 0.7³ (= 0.343)
(b) A1 0.784 cao (accept awrt 0.784)
(b) Answer: 0.784

Question 18

(a) M1 sets up n/(n+4) x (n-1)/(n+3) = 1/3
(a) M1 correctly clears fractions: 3n(n-1) = (n+4)(n+3)
(a) A1 correctly expands and simplifies to n² - 5n - 6 = 0 cso
(a) Answer: n² - 5n - 6 = 0
(b) M1 factorises (n - 6)(n + 1) = 0, or uses the quadratic formula
(b) A1 n = 6 cao, rejecting n = -1 since a negative number of pens is not possible
(b) Answer: n = 6

Question 19

(a) M1 P(biased) x 0.9 + P(fair) x 0.5, with P(biased) = 1/4 and P(fair) = 3/4
(a) M1 = 0.25 x 0.9 + 0.75 x 0.5 (= 0.225 + 0.375)
(a) A1 0.6 cao
(a) Answer: 0.6
(b) M1 recognises P(biased and heads) = 0.225, ft from part (a)
(b) M1 divides by P(heads): 0.225 / 0.6, ft from part (a)
(b) A1 0.375 oe (3/8) cao
(b) Answer: 3/8

Probability: Fluency and Exam Drill (Part 2) - Worksheets, Questions and Revision

More on this topic

Next in this strand

2.12DB Probability: Fluency and Exam Drill (Part 2)

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Question 17

Question 18

Question 19

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Question 17

Question 18

Question 19